Newspace parameters
| Level: | \( N \) | \(=\) | \( 440 = 2^{3} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 440.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(70.5688807177\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 637x^{3} + 2294x^{2} + 68016x - 243936 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{5} - 2x^{4} - 637x^{3} + 2294x^{2} + 68016x - 243936 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 42\nu^{3} + 1129\nu^{2} - 17046\nu - 150720 ) / 2172 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 42\nu^{3} - 43\nu^{2} + 19218\nu - 128382 ) / 1086 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15\nu^{4} + 94\nu^{3} - 8247\nu^{2} - 27394\nu + 581844 ) / 2172 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} - 2\beta _1 + 257 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} - 12\beta_{3} + 21\beta_{2} + 415\beta _1 - 765 \)
|
| \(\nu^{4}\) | \(=\) |
\( 126\beta_{4} + 625\beta_{3} + 968\beta_{2} - 1874\beta _1 + 107303 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −21.0364 | 0 | 25.0000 | 0 | −86.5420 | 0 | 199.529 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −12.1463 | 0 | 25.0000 | 0 | 190.908 | 0 | −95.4669 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.58009 | 0 | 25.0000 | 0 | −221.025 | 0 | −230.183 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 11.4276 | 0 | 25.0000 | 0 | −43.2125 | 0 | −112.410 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 23.3352 | 0 | 25.0000 | 0 | −90.1288 | 0 | 301.531 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 440.6.a.b | ✓ | 5 |
| 4.b | odd | 2 | 1 | 880.6.a.s | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 440.6.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 880.6.a.s | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 2T_{3}^{4} - 637T_{3}^{3} - 2294T_{3}^{2} + 68016T_{3} + 243936 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(440))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 2 T^{4} + \cdots + 243936 \)
|
| $5$ |
\( (T - 25)^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 14222172672 \)
|
| $11$ |
\( (T - 121)^{5} \)
|
| $13$ |
\( T^{5} + \cdots + 10655605118976 \)
|
| $17$ |
\( T^{5} + \cdots + 29\!\cdots\!36 \)
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| $19$ |
\( T^{5} + \cdots + 14253622702272 \)
|
| $23$ |
\( T^{5} + \cdots - 38\!\cdots\!56 \)
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| $29$ |
\( T^{5} + \cdots - 73\!\cdots\!36 \)
|
| $31$ |
\( T^{5} + \cdots - 95\!\cdots\!28 \)
|
| $37$ |
\( T^{5} + \cdots - 20\!\cdots\!44 \)
|
| $41$ |
\( T^{5} + \cdots + 51\!\cdots\!44 \)
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| $43$ |
\( T^{5} + \cdots + 24\!\cdots\!68 \)
|
| $47$ |
\( T^{5} + \cdots + 58\!\cdots\!60 \)
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| $53$ |
\( T^{5} + \cdots - 70\!\cdots\!60 \)
|
| $59$ |
\( T^{5} + \cdots - 22\!\cdots\!28 \)
|
| $61$ |
\( T^{5} + \cdots - 96\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots + 14\!\cdots\!64 \)
|
| $71$ |
\( T^{5} + \cdots - 57\!\cdots\!28 \)
|
| $73$ |
\( T^{5} + \cdots - 69\!\cdots\!92 \)
|
| $79$ |
\( T^{5} + \cdots + 11\!\cdots\!44 \)
|
| $83$ |
\( T^{5} + \cdots + 22\!\cdots\!56 \)
|
| $89$ |
\( T^{5} + \cdots - 13\!\cdots\!84 \)
|
| $97$ |
\( T^{5} + \cdots + 14\!\cdots\!28 \)
|
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